A snowboarder glides down a 50-m-long, 15 degree hill. She then glides horizontally for 10 m before reaching a 25 degree upward slope. Assume the snow is frictionless. (b) How far can she travel up the 25 degree slope?

Video transcript

Hey, everyone in this problem, a toy car track consists excessively of a 30 centimeter straight inclined downhill section making an angle of 10 degrees with the horizontal, A 70 Centim horizontal section in a 50 centimeter uphill section making an angle of 20 degrees with the horizontal, all the sections are frictionless. A car is released from rest at the top of the 30 Centim downhill. And we're asked, what is the distance traveled by the car Along the 20° uphill? We're given four answer choices. A 13.6 centimeters. B 15.2 centimeters C 26 cm and d 30.4 cm. Let's go ahead and draw this track. Oh, now we have the first part of the track and it's an incline and we know that this incline makes a oops serious. Let's try that again. The incline makes a 10° angle with the horizontal And the incline is 30 cm long. And we're gonna start with our car. I'm just gonna draw the car as a circle and it's going to be at rest. OK? It's gonna be released from rest. So its initial speed is 0m/s and it's gonna travel down this incline. Once it gets to the bottom of the incline, we have a horizontal section that is 70 cm long. And then we have an uphill section That makes an angle of 20° with the horizontal. And what we wanna figure out is the distance traveled by the car on the 20° uphill. So at some point, this car is going to stop and come back down. And so the final speed here is going to be zero as well. The car is gonna be momentarily at rest when it reaches its maximum height and then starts to come back down and we wanna find this distance. Delta X. So this problem has three stages, ok? We have the first stage where we have the downhill incline. We have the second stage where we have this horizontal piece, we're gonna call the first downhill incline. Stage zero. This horizontal is gonna be stage one and then the portion that goes uphill Is going to be stage two. Yes. So we have these three stages to consider in our motion. So let's write out all the information we know for each of the stages. OK? So for Stage zero we wanna have a tilted axis if we use the tilted axis and the motion is just going to be a longer axis. And we don't have to worry about breaking the motion into two components, the X and Y direction. OK. So we're gonna have a tilted axis. The initial speed is going to be zero m/s. We don't know what the final speed will be. We know that we're gonna travel a distance delta X of cm. And we want to convert this to our standard units to so to go from 30 cm into meters, we're going to divide by 100 and we get 0. m in our acceleration. Well, this is gonna accelerate because of gravity, right, the car is not going straight down but it is going down along that incline. So gravity is going to affect it. So if we draw out are tilted axis here and we have our car, we imagine we have gravity pointing straight downwards, we want to project that gravity onto the tilted axis we're using and the acceleration is going to be pointing down along our axis in the direction of motion. And we can take the direction of motion to be positive. We know that the angle between the gravity and the perpendicular to our axis is going to be the same angle As the axis with the horizontal, which is 10°. And so we can find the component of the acceleration in our direction of motion. And that is gonna be sine of the angle 10° multiplied by that gravitational acceleration G. And so this is equal to sine of 10° Multiplied by 9.8 m/s squared. All right. So that's stage zero. What about Stage 1? Oh at stage one it's just horizontal means we have no acceleration due to gravity acting on it. We have no friction. We're told that this is a frictionless section. So the initial speed in stage one Is going to be equal to the final speed of stage one and this is also gonna be equal to the final stage, the final speed of stage one or stage zero as well. Sorry. Ok. So that final speed that the car is at when it gets to the bottom of the incline is gonna be the same speed it stays at throughout this entire horizontal section. OK? So we stage two or stage one, we don't have to worry about the acceleration or anything like that When we look at stage two, which is what we're really interested in because this is the distance we're looking for. We have the initial speed we don't know, but it is going to be the exact same as the final speed from stage one, Which we know is equal to the final stage from stage or the final speed from stage zero via We know that the final speed will be 0m per second because that car is going to stop momentarily at its maximum height before it comes back downwards. So the final speed is going to be zero. We're looking for Delta X and we're gonna call it Delta X two. Same with V F. We're gonna add a subscript to there To indicate stage two in our acceleration, we can break it up just like we did for stage zero. So we have our incline, we have our car and gravity is going to act straight downwards. We project to that on to our incline. We're gonna take positive to be the direction of uphill motion. Then the acceleration is going to be acting in the negative direction. And the angle we have is going to be 20° between the gravity and the perpendicular to our plane. OK. So the acceleration is going to be negative because it's acting downwards in the opposite direction of motion. The opposite direction is we've chosen as our positive direction, Stine of 20° multiplied by the gravitational acceleration 9.8 m/s squared. All right, we aren't given any time information about the time for stage two. So we wanna find delta X two, We only have two known values when we're looking at the stage two information, we know VF two and we know a two, but we don't know the initial speed or the time. If we can find one more known value, we'll be able to use our kinematic equations to solve our Delta X two. So we don't know we not, but we know that it's equal to the final speed from stage zero the initial stage. So if we go back to our initial stage, OK. We have three known values. We can use those to find V F and then we can use that V F value as our initial speed for sage two and soft for delta X two. OK. So we're gonna kind of start at the beginning and work our way up to this stage two pro. So let's give ourselves some more room. And we're gonna start with stage zero that initial stage and we want to solve for the final speed. Now, we're gonna take the kinematic equation that doesn't include time because we don't have information about that. And that's not what we're looking for. We have V F squared is equal to V knot squared Plus two multiplied by a multiplied by Delta X V F squared is equal to V is zero. So this entire first term goes to zero, we get two multiplied by the acceleration sine 10 degrees multiplied by 9.8 m per second squared, multiplied by delta X 0.3 m. If we work this out on our calculator, we have that V F squared is equal to 1.02, 105 meter squared per second squared. We take the square route. OK? And we're gonna take the positive route because we know that this is traveling in the positive direction that we've chosen. OK. So it's gonna have a positive velocity, positive speed. So we have the square root is going to give us 01. m/s. And that is that final speed when we reach the bottom of that downward incline. All right. So V F is this 1.1047 m per second. We know that that means that the entire speed, the speed during the entire stage one, that entire horizontal section is also going to be V F equals 1.1047 m per second. So that is gonna be our initial speed when we get to the uphill ramp. Now, for the uphill ramp, we want to take the same equation. We don't have information about the time. T that's not what we're looking for. So we're gonna take the same equation V F squared. And in this case, it's V F two squared is equal to V, not two squared Plus two multiplied by a two multiplied by Delta X two. Now, in this case, we know VNA and V F, what we're looking for is delta X two substituting in the values we have zero on the left hand side is equal to our initial speed, which is the same as the final speed of stage 1. m per second squared plus two, multiplied by the acceleration, negative sign 20 degrees multiplied. Let me write this down below. So we have room I don't wanna squeeze it in there. So V F squared is going to give us zero. So we have zero is equal to the initial speed. 1.1047 m per second squared plus two, multiplied by negative sign 20 degrees, multiplied by 9.8 m per second squared Multiplied by Delta X two. And that value of Delta X two is what we are looking for. Now, if we simplify on the right hand side, we have 1.1047 m per second squared, we know that value because we took the square root of that earlier. We have 1.2105 m squared per second squared -6.703, 6 m per second squared, multiplied by Delta X two. And if we solve for delta X two, OK, we can move the constant term to the left hand side and then divide by 6.7036 m per second squared. We get a delta X two value of 0. 23 m and that is a distance up that incline that the car travels. Ok? Now this is in meters and our answer choices are in centimeters. So what we wanna do is convert this back into centimeters in order to do that, we are going to multiply by 100 we get that delta X two is equal to 15.23 centimeters. Looking back at our answer choices And rounding to three significant digits we found the distance traveled by the car along a 20° uphill is 15.2 cm which corresponds with answer choice. B thanks everyone for watching. I hope this video helped you in the next one.